(Please read part 1 first)

Now that you have some idea as to what functions and limits are, we can explore the derivative itself.

From algebra, you learned that you can use two points to calculate the slope of a line (y2 - y1)/(x2 - x1) where (x1, y1) and (x2, y2) are the two points you know. However, this process only calculates the average slope of the line, not the instantaneous slope.

For example, look at f(x) = x^2. Say I wanted to find the slope at f(1). Well, f(1) = 1^2 = 1, so we have the point (1, 1). So, we need a second point. Let's take x = 5, so f(5) = 5^2 = 25, so we have the point (5, 25). Thus, the slope is (25 - 1) / (5 - 1) = 24 / 4 = 6. But, if we take a look at f(x) = x^2 on a graphing calculator (I so wish I could do drawings in this blog), a slope of 6 is far too steep to be the slope at x = 1. What happened is that we found the average slope instead of the instant slope.

Now, say we try a closer point, taking x = 2. Then, f(2) = 2^2 = 4, giving us (2, 4) as our point. Then, the slope is (4 - 1) / (2 -1) = 3, which looks better but still is too steep. Trying even closer, at x = 1.1 gives us f(1.1) = 1.1^2 = 1.21, so we have (1.1, 1.21). The slope is (1.21 - 1) / (1.1 - 1) = 0.21 / 0.1 = 2.1, much closer to the slope. The closer we make the second point to 1, the better our estimate of the slope becomes. This means that if the second point was actually the same as our first point, we would have the actual instantaneous slope, but we cannot do that, since (1 - 1) / (1 - 1) = 0/0, which is divide by zero. As I implied in part 1 though, we can use a limit to sometimes find what is going on in a divide by zero case, which is precisely what the derivative is...